Using the Mann-Whitney U Test The Mann-Whitney U test is equivalent to the Wilcoxon rank-sum test for independent samples in the sense that they both apply to the same situations and always lead to the same conclusions. In the Mann-Whitney U test we calculate

\(z = \frac{{U - \frac{{{n_1}{n_2}}}{2}}}{{\sqrt {\frac{{{n_1}{n_2}\left( {{n_1} + {n_2} + 1} \right)}}{{12}}} }}\)

Where

\(U = {n_1}{n_2} + \frac{{{n_1}\left( {{n_1} + 1} \right)}}{2} - R\)

and R is the sum of the ranks for Sample 1. Use the student course evaluation ratings in Table 13-5 on page 621 to find the z test statistic for the Mann-Whitney U test. Compare this value to the z test statistic found using the Wilcoxon rank-sum test.

The Wilcoxon rank-sum test and the Mann-Whitney U test arenon-parametric tests used to test the difference of medians between two independent samples when both tests are applied to similar situations.

The data are given on course evaluation ratings for courses taught by male and female professors.

The ranks of the observations from the two samples are given using the following steps:

• Combine the two samples and label each observation with the sample name/number it comes from.
• The smallest observation is assigned rank 1; the next smallest observation is assigned rank 2, and so on until the largest value.
• If two observations have the same value, the mean of the ranks is assigned to them.

The following table shows the ranks:

\(\)

The sum of the ranks corresponding to female professors is equal to:

\(\begin{array}{c}R = 20.5 + 20.5 + 23.5 + .... + 3\\ = 159.5\end{array}\)

Let\({n_1}\)be the sample size corresponding to the female professors.

Let\({n_2}\)be the sample size corresponding to the male professors.

Here,

\(\begin{array}{l}{n_1} = 12\\{n_2} = 15\end{array}\)

The value of U is computed as follows:

\(\begin{array}{c}U = {n_1}{n_2} + \frac{{{n_1}\left( {{n_1} + 1} \right)}}{2} - R\\ = \left( {12 \times 15} \right) + \frac{{12\left( {12 + 1} \right)}}{2} - 159.5\\ = 98.5\\\end{array}\)

The value of the z statistic for the Mann Whitney U test is as follows:

\(\begin{array}{c}z = \frac{{U - \frac{{{n_1}{n_2}}}{2}}}{{\sqrt {\frac{{{n_1}{n_2}\left( {{n_1} + {n_2} + 1} \right)}}{{12}}} }}\\ = \frac{{98.5 - \frac{{12 \times 15}}{2}}}{{\sqrt {\frac{{\left( {12 \times 15} \right)\left( {12 + 15 + 1} \right)}}{{12}}} }}\\ = 0.41\end{array}\)

Thus, z is equal to 0.41.

The mean value\(\left( {{\mu _R}} \right)\)is as follows:

\(\begin{array}{c}{\mu _R} = \frac{{{n_1}\left( {{n_1} + {n_2} + 1} \right)}}{2}\\ = \frac{{12\left( {12 + 15 + 1} \right)}}{2}\\ = 168\end{array}\)

The standard deviation\(\left( {{\sigma _R}} \right)\)is as follows:

\(\begin{array}{c}{\sigma _R} = \sqrt {\frac{{{n_1}{n_2}\left( {{n_1} + {n_2} + 1} \right)}}{{12}}} \\ = \sqrt {\frac{{12 \times 15\left( {12 + 15 + 1} \right)}}{{12}}} \\ = 20.49\end{array}\)

The test statistic value for the Wilcoxon rank-sum test is calculated as follows:

\(\begin{array}{c}z = \frac{{R - {\mu _R}}}{{{\sigma _R}}}\; \sim N\left( {0,1} \right)\\ = \frac{{159.5 - 168}}{{20.49}}\\ = - 0.41\end{array}\)

Thus, z is equal to -0.41.

The values of the test statistic (z) arethe same for both the tests, i.e.,equal to 0.41,but the signs are opposite.

The value of z for the Mann Whitney U test is positive, whereas the value of z for the Wilcoxon rank-sum test is negative.